Generalized permutation matrices and non-weight modules over $\mathfrak{sl}(m|1)$

TL;DR

This paper investigates a specific category of modules over the Lie superalgebra rak{sl}(m|1), constructing explicit modules, proving an isomorphism theorem, and establishing criteria for indecomposability.

Contribution

It introduces a new family of rak{h}-free modules over rak{sl}(m|1), along with an isomorphism theorem and indecomposability criteria.

Findings

Constructed explicit rak{h}-free modules of rank (k|k).

Proved an isomorphism theorem for these modules.

Established criteria for indecomposability.

Abstract

We study the category $\mathcal{M}_{\mathfrak{sl}(m|1)}(k|k)$ of $\mathcal U(\mathfrak h)\text{-free}$ $\mathcal U(\mathfrak{sl}(m|1))$-modules of rank $k$ in each parity (rank $(k|k)$), where $k\in\mathbb{Z}_{\geq1}$. We construct an explicit family of such modules, provide an isomorphism theorem, and establish an indecomposability criterion.